1. LU Decomposition http://numericalmethods.eng.usf.edu
Chemical Engineering Majors
Authors: Autar Kaw
Transforming Numerical Methods Education for STEM Undergraduates
11/29/2018

2. LU Decomposition http://numericalmethods.eng.usf.edu

3. LU Decomposition
LU Decomposition is another method to solve a set of simultaneous linear equations
Which is better, Gauss Elimination or LU Decomposition?
To answer this, a closer look at LU decomposition is needed.

4. LU Decomposition Method [A] = [L][U] where
For most non-singular matrix [A] that one could conduct Naïve Gauss Elimination forward elimination steps, one can always write it as
[A] = [L][U]
where
[L] = lower triangular matrix
[U] = upper triangular matrix

5. How does LU Decomposition work?
If solving a set of linear equations
[A][X] = [C]
If [A] = [L][U] then
[L][U][X] = [C]
Multiply by
[L]-1
Which gives
[L]-1[L][U][X] = [L]-1[C]
Remember [L]-1[L] = [I] which leads to
[I][U][X] = [L]-1[C]
Now, if [I][U] = [U] then
[U][X] = [L]-1[C]
Now, let
[L]-1[C]=[Z]
Which ends with
[L][Z] = [C] (1)
and
[U][X] = [Z] (2)

6. LU Decomposition How can this be used? Given [A][X] = [C]
Decompose [A] into [L] and [U]
Solve [L][Z] = [C] for [Z]
Solve [U][X] = [Z] for [X]

7. When is LU Decomposition better than Gaussian Elimination?
To solve [A][X] = [B]
Table. Time taken by methods
where T = clock cycle time and n = size of the matrix
So both methods are equally efficient.
Gaussian Elimination
LU Decomposition

8. Time taken by Gaussian Elimination Time taken by LU Decomposition
To find inverse of [A]
Time taken by Gaussian Elimination Time taken by LU Decomposition
Table 1 Comparing computational times of finding inverse of a matrix using LU decomposition and Gaussian elimination. n 10
100
1000
10000
CT|inverse GE / CT|inverse LU
3.28
25.83
250.8
2501

9. Method: [A] Decompose to [L] and [U]
[U] is the same as the coefficient matrix at the end of the forward elimination step.
[L] is obtained using the multipliers that were used in the forward elimination process

10. Using the Forward Elimination Procedure of Gauss Elimination
Finding the [U] matrix
Using the Forward Elimination Procedure of Gauss Elimination
Step 1:

11. Finding the [U] Matrix Matrix after Step 1: Step 2:

12. Using the multipliers used during the Forward Elimination Procedure
Finding the [L] matrix
Using the multipliers used during the Forward Elimination Procedure
From the first step of forward elimination

13. Finding the [L] Matrix From the second step of forward elimination

14. Does [L][U] = [A]? ?

15. Example: Liquid-Liquid Extraction
A liquid-liquid extraction process conducted in the Electrochemical Materials Laboratory involved the extraction of nickel from the aqueous phase into an organic phase. A typical set of experimental data from the laboratory is given below:
Ni aqueous phase, a (g/l) 2
2.5 3
Ni organic phase, g (g/l)
8.57 10 12
Assuming g is the amount of Ni in organic phase and a is the amount of Ni in the aqueous phase, the quadratic interpolant that estimates g is given by:

16. Example: Liquid-Liquid Extraction
The solution for the unknowns x1, x2, and x3 is given by
Find the values of x1, x2, and x3 using LU Decomposition. Estimate the amount of nickel in organic phase when 2.3 g/l is in the aqueous phase using quadratic interpolation

17. Example: Liquid-Liquid Extraction
Use Forward Elimination to find the [U] matrix
Step 1

18. Example: Liquid-Liquid Extraction
The matrix after the 1st step:
Step 2

19. Example: Liquid-Liquid Extraction
Find the [L] matrix
Use the multipliers from the Forward Elimination Procedure
From the first step of forward elimination
From the second step of forward elimination

20. Example: Liquid-Liquid Extraction
Does [L][U] = [A]?

21. Example: Liquid-Liquid Extraction
Set [L][Z] = [C]
Solve for [Z]

22. Example: Liquid-Liquid Extraction
Complete the forward substitution to solve for [Z]

23. Example: Liquid-Liquid Extraction
Set [U][X] = [Z]
Solve for [X]
The 3 equations become

24. Example: Liquid-Liquid Extraction
Solve for [X]

25. Example: Liquid-Liquid Extraction
The Solution Vector is:

26. Example: Liquid-Liquid Extraction
The polynomial that passes through the three data points is then
Solution
Where g is grams of nickel in the organic phase and a is the grams/liter in the aqueous phase.
When 2.3g/l is in the aqueous phase, using quadratic interpolation, the estimated amount of nickel in the organic phase is

27. Finding the inverse of a square matrix
The inverse [B] of a square matrix [A] is defined as
[A][B] = [I] = [B][A]

28. Finding the inverse of a square matrix
How can LU Decomposition be used to find the inverse?
Assume the first column of [B] to be [b11 b12 … bn1]T
Using this and the definition of matrix multiplication
First column of [B] Second column of [B]
The remaining columns in [B] can be found in the same manner

29. Example: Inverse of a Matrix
Find the inverse of a square matrix [A]
Using the decomposition procedure, the [L] and [U] matrices are found to be

30. Example: Inverse of a Matrix
Solving for the each column of [B] requires two steps
Solve [L] [Z] = [C] for [Z]
Solve [U] [X] = [Z] for [X]
Step 1:
This generates the equations:

31. Example: Inverse of a Matrix
Solving for [Z]

32. Example: Inverse of a Matrix
Solving [U][X] = [Z] for [X]

33. Example: Inverse of a Matrix
Using Backward Substitution
So the first column of the inverse of [A] is:

34. Example: Inverse of a Matrix
Repeating for the second and third columns of the inverse
Second Column Third Column

35. Example: Inverse of a Matrix
The inverse of [A] is
To check your work do the following operation
[A][A]-1 = [I] = [A]-1[A]

36. Additional Resources
For all resources on this topic such as digital audiovisual lectures, primers, textbook chapters, multiple-choice tests, worksheets in MATLAB, MATHEMATICA, MathCad and MAPLE, blogs, related physical problems, please visit

37. THE END